Continuity mistake: Batman's utility belt is normal in the beginning, but suddenly has a triple sized compartment when Batman goes to get up on the catwalk with his rope. (00:03:30)
Continuity mistake: When Batman and Robin scale the warehouse wall then watch Joker and his cohorts, each of the two windows have latches at the bottom, but when the Dynamic Duo burst in through the window, only the left window has a latch. (00:11:45)
Continuity mistake: When Batman and Robin crash through the window, there's no glass left in the frame - at least not until the next shot, when several large shards that weren't there before suddenly appear at the edges. (00:11:50)
Continuity mistake: During the climactic fight, Batman tosses the Joker utility belt, causing confetti and streamers to fall down all over the place in some shots, while there is no such paper rain in others. (00:21:40)
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon